I understand that vectors are a subset of Tensors. Can someone please define tensors in a simple manner? I read about tensors from quora and saw some videos too, but still couldn't grasp its inner meaning!
Hi Suhas,
the simplest definition I know of is: While multiplication with a scalar affects the length of a vector, multiplication with a tensor affects both its length and its direction.
Nearest example in electromagnetics: Permittivity and permeability of isotropic media are scalars, so the directions of E and D are the same, and so are the directions of H and B. In anisotropic media, the directions can be different, and as a consequence, permittivity and / or permeability have to be represented by tensors. In this case tensors are just another name for 3 by 3 matrices.
Quite another matter is the representation of the combined E and B fields by tensor notation. Personally, a vector equation split into three equations for the three spatial components (like Maxwell's original equations) looks much more cumbersome to me than usual vector notation, but the even more compact form of tensor notation looks equally cumbersome! I guess it's just a matter of practice and familiarity, and I seem to remember reading somewhere that Maxwell wrote component equations in spite of his knowledge of quaternions, in order to ease the matter for his contemporaries.
Sir @Joerg Fricke,
So the 3X3 matrix defined by
epsilonExx epsilonExy epsilonExz
epsilonEyx epsilonEyy epsilonEyz
epsilonEzx epsilonEzy epsilonEzz
Great explanation. I understand to a certain extent now!
Thus, a Scalar in an anisotropic medium can be defined only by a Tensor, right sir?!
I would say: An anisotropic quantity in a medium can be defined only by a tensor.
Some days ago, Raymond Rumpf sent me a link to his lectures on CEM (slides in pdf and recording of lectures in mp4). Around slide 24 in Lecture 2 he talks about anisotropic permittivity and permeability:
http://emlab.utep.edu/ee5390cem.htm
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